{"id":"n-a9609862a68b4355","revision":0,"first_index":0,"fingerprint":"f6-46d787fbc997ccf4","name":"shift_x_plus_one of basis_map_A013609 of A Jacobsthal triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","2"],["15","16","4"],["63","102","52","8"],["279","588","448","144","16"],["1215","3186","3240","1584","368","32"],["5319","16632","21132","13888","4944","896","64"],["23247","84510","128844","106392","51152","14240","2112","128"],["101655","421092","748872","744720","451456","170176","38784","4864","256"],["444447","2066634","4199904","4885632","3576288","1703488","526336","101376","11008","512"],["1943271","10021968","22910580","30520512","26187744","15089920","5899392","1541120","256768","24576","1024"],["8496495","48130038","122241636","183496104","180604512","122171328","57849984","19135744","4324096","634368","54272","2048"],["37149111","229287996","640536336","1069736976","1188014256","923013504","513648640","205951488","58948864","11725824","1536000","118784","4096"],["162426303","1084928418","3306154968","6080226768","7521128208","6600515040","4222758144","1991366144","691237120","174177792","30922752","3657728","258048","8192"],["710173575","5103962280","16848691740","33833557440","46130871600","45133939584","32643250368","17709918208","7233942784","2211158016","497243136","79675392","8589312","557056","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","225","1475","9625","62875","410625","2681875","17515625","114396875","747140625","4879671875","31869765625","208145546875"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["14","4"],["-8","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["33","28","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-f068516af8b6b455","recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176309,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-046492379129f959"]},{"id":176310,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-046492379129f959","e-85d481efeacf630e"]},{"id":176311,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-046492379129f959","e-85d481efeacf630e"]},{"id":176312,"position":3,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-85d481efeacf630e"]},{"id":176313,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-85d481efeacf630e"]}],"region_exact":{},"incoming_edges":[{"id":"e-85d481efeacf630e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-a9609862a68b4355","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"0c0dc845abd447bf2bc8b930abdae26dae116f5e21016accc46d97f1b8cc1d74","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-546dca52670dddd1","role":"sequence","name":"basis_map_A013609 of A Jacobsthal triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-f068516af8b6b455","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-a9609862a68b4355","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"e440290dd8ed2e5fb5b5fab4436278b98d6a0eb00aef41493b801fa8c15f5b29","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-046492379129f959","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a9609862a68b4355","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"436915172aeeda2fccd1d483e6656109b951a8333c52d05c754cefd78b61104a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-546dca52670dddd1","role":"sequence","name":"basis_map_A013609 of A Jacobsthal triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      3,       2;\n  2:     15,      16,       4;\n  3:     63,     102,      52,       8;\n  4:    279,     588,     448,     144,      16;\n  5:   1215,    3186,    3240,    1584,     368,      32;\n  6:   5319,   16632,   21132,   13888,    4944,     896,     64;\n  7:  23247,   84510,  128844,  106392,   51152,   14240,   2112,    128;\n  8: 101655,  421092,  748872,  744720,  451456,  170176,  38784,   4864,   256;\n  9: 444447, 2066634, 4199904, 4885632, 3576288, 1703488, 526336, 101376, 11008, 512;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = -(8*n-14-4*x)*P(n-1,x) + (33+28*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 14 - 4*x)*P[n - 1, x] + (33 + 28*x + 4*x^2)*D[P[n - 1, x], x])/(2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n - 14 - 4*x)*v[i - 1] + (33 + 28*x + 4*x^2)*deriv(v[i - 1], x))/(2*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba196dcd01800741","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-34478b6712498a37","e-046492379129f959"],"text":"A113953 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A113953 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}